Find one value of $x$ that is a solution to the equation: $(3x-2)^2-4=9x-6$ $x=$
We could solve for $x$ by expanding $(3x-2)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $9x-6=3({3x-2})$. This means that we can rewrite the equation as: $({3x-2})^2-4=3({3x-2})$ If we let ${p}={3x-2}$, we can see that this equation is in the form: ${p}^2-4=3{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2-4&=3{p}\\\\ {p}^2-3{p}-4&=0\\\\ ({p}-4)({p}+1)&=0\\\\ {p}=4\ &\text{or} \ \ {p}=-1 \end{aligned}$ Since ${p}={3x-2}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${3x-2}=4\ \ \ \text{or} \ \ \ {3x-2}=-1$ When we solve ${3x-2}=4$, we find that $x=2$. When we solve ${3x-2}=-1$, we find that $x=\dfrac{1}{3}$. In conclusion, the two solutions of the equation $(3x-2)^2-4=9x-6$ are $x=2$ and $x=\dfrac{1}{3}$. [Is there another way to solve for x?]